Groupoid
Encyclopedia
In mathematics
, especially in category theory
and homotopy theory, a groupoid (less often Brandt groupoid or virtual group) generalises the notion of group
in several equivalent ways. A groupoid can be seen as a:
Special cases include:
Groupoids are often used to reason about geometrical objects such as manifold
s. Heinrich Brandt
introduced groupoids implicitly via Brandt semigroups in 1926.
and a partial function
Here * is not a binary operation
because it is not necessarily defined for all possible pairs of G-elements. The precise conditions under which * is defined are not articulated here and vary by situation.
and −1 have the following axiomatic properties. Let a, b, and c be elements of G. Then:
In short:
From these axioms, two easy and convenient properties follow:
Proof of first property: from 3. we obtain (a−1)−1=(a−1)−1 * a−1 * a = a. ✓
Proof of second property: since a * b and b * b−1 are defined, so is a * b * b−1=a. Therefore a * b * b−1 * a−1 is also defined. From 3. we obtain (a * b)−1 = (a * b)−1 * a*b * b−1 * a−1 = b−1 * a−1. ✓
in which every morphism
is an isomorphism
, and hence invertible. More precisely, a groupoid G is:
The objects and morphisms have the properties:
Moreover, if f : x → y, g : y → z, and h : z → w, then:
If f is an element of G(x,y) then x is called the source of f, written s(f), and y the target of f (written t(f)).
of all of the sets G(x,y) (i.e. the sets of morphisms from x to y). Then
and 
become partially defined operations on G, and 
will in fact be defined everywhere; so we define * to be 
and 
to be 
. Thus we have a groupoid in the algebraic sense. Explicit reference to G0 (and hence to 
) can be dropped.
Conversely, given a groupoid G in the algebraic sense, let G0 be the set of all elements of the form x * x−1 with x varying through G and define G(x*x -1,y*y -1) as the set of all elements f such that y * y -1 * f * x * x -1 exists. Given f∈G(x*x-1,y*y -1) and g∈G(y*y -1,z*z -1), their composite is defined as g * f ∈ G(x*x -1,z*z -1). To see this is well defined, observe that since z*z -1 * g * y*y -1 and y*y-1 * f * x*x-1 exist, so does z*z-1 * g * y*y-1 * y*y-1 * f * x*x -1 = z*z-1 * g*f * x*x -1. The identity morphism on x*x−1 is then x*x−1 itself, and the category-theoretic inverse of f is f-1.
Sets in the definitions above may be replaced with class
es, as is generally the case in category theory
.
that is itself a groupoid. A groupoid morphism is simply a functor between two (category-theoretic) groupoids. The category whose objects are groupoids and whose morphisms are groupoid morphisms is called the groupoid category, or the category of groupoids, denoted Grpd.
Particular kinds of morphisms of groupoids are of interest. A morphism $p: E \to B$ of groupoids is called a fibration if for each object $x$ of $E$ and each morphism $b$ of $B$ starting at $p(x)$ there is a morphism $e$ of $E$ starting at $x$ such that $p(e)=b$. A fibration is called a covering morphism if further such an $e$ is unique. The covering morphisms of groupoids are especially useful because they can be used to model covering map
s of spaces. It is also true that the category of covering morphisms of a given groupoid $B$ is equivalent to the category of actions of the groupoid $B$ on sets.
whose entries range over K. Matrix multiplication
interprets composition. If G = GL*(K), then the set of natural number
s is a proper subset of G0, since for each natural number
n, there is a corresponding identity matrix
of dimension n. G(m,n) is empty
unless m=n, in which case it is the set of all nxn invertible matrices.
X, let G0 be the set X. The morphisms from the point p to the point q are equivalence classes of continuous path
s from p to q, with two paths being equivalent if they are homotopic.
Two such morphisms are composed by first following the first path, then the second; the homotopy equivalence guarantees that this composition is associative. This groupoid is called the fundamental groupoid of X, denoted
(X). The usual fundamental group 
is then the vertex group for the point x.
An important extension of this idea is to consider the fundamental groupoid
(X,A) where A is a set of "base points" and a subset of X. Here, one considers only paths whose endpoints belong to A. 
(X,A) is a sub-groupoid of 
(X). The set A may be chosen according to the geometry of the situation at hand.
denoted by infix
, then a groupoid "representing" this equivalence relation can be formed as follows:
G acts on the set X, then we can form the action groupoid representing this group action
as follows:
More explicitly, the action groupoid is the set
with source and target maps s(g,x) = x and t(g,x) = gx. It is often denoted 
(or 
). Multiplication (or composition) in the groupoid is then 
which is defined provided y=gx.
For x in X, the vertex group consists of those (g,x) with gx = x, which is just the isotropy subgroup at x for the given action (which is why vertex groups are also called isotropy groups).
Another way to describe G-sets is the functor category
, where 
is the groupoid (category) with one element and isomorphic
to the group G. Indeed, every functor F of this category defines a set X=F
and for every g in G (i.e. for every morphism in 
) induces a bijection
Fg : X→X. The categorical structure of the functor F assures us that F defines a G-action on the set X. The (unique) representable functor
F :
→
is the Cayley representation
of G. In fact, this functor is isomorphic to
and so sends 
to the set 
which is by definition the "set" G and the morphism g of 
(i.e. the element g of G) to the permutation Fg of the set G. We deduce from the Yoneda embedding that the group G is isomorphic to the group {Fg | g∈G}, a subgroup
of the group of permutation
s of G.
generalize to groupoids, with the notion of functor
replacing that of group homomorphism
.
If x is an object of the groupoid G, then the set of all morphisms from x to x forms a group G(x). If there is a morphism f from x to y, then the groups G(x) and G(y) are isomorphic
, with an isomorphism given by the mapping
g → fgf −1.
Every connected groupoid (that is, one in which any two objects are connected by at least one morphism) is isomorphic to a groupoid of the following form. Pick a group G and a set (or class) X. Let the objects of the groupoid be the elements of X. For elements x and y of X, let the set of morphisms from x to y be G. Composition of morphisms is the group operation of G. If the groupoid is not connected, then it is isomorphic to a disjoint union
of groupoids of the above type (possibly with different groups G for each connected component). Thus any groupoid may be given (up to
isomorphism) by a set of ordered pair
s (X,G).
Note that the isomorphism described above is not unique, and there is no natural choice. Choosing such an isomorphism for a connected groupoid essentially amounts to picking one object x0, a group isomorphism
h from G(x0) to G, and for each x other than x0, a morphism in G from x0 to x.
In category-theoretic terms, each connected component of a groupoid is equivalent (but not isomorphic) to a groupoid with a single object, that is, a single group. Thus any groupoid is equivalent to a multiset
of unrelated groups. In other words, for equivalence instead of isomorphism, one need not specify the sets X, only the groups G.
Consider the examples in the previous section. The general linear groupoid is both equivalent and isomorphic to the disjoint union of the various general linear group
s GLn(F). On the other hand:
The collapse of a groupoid into a mere collection of groups loses some information, even from a category-theoretic point of view, because it is not natural. Thus when groupoids arise in terms of other structures, as in the above examples, it can be helpful to maintain the full groupoid. Otherwise, one must choose a way to view each G(x) in terms of a single group, and this choice can be arbitrary. In our example from topology
, you would have to make a coherent choice of paths (or equivalence classes of paths) from each point p to each point q in the same path-connected component.
As a more illuminating example, the classification of groupoids with one endomorphism
does not reduce to purely group theoretic considerations. This is analogous to the fact that the classification of vector space
s with one endomorphism is nontrivial.
Morphisms of groupoids come in more kinds than those of groups: we have, for example, fibration
s, covering morphisms, universal morphisms, and quotient morphisms. Thus a subgroup H of a group G yields an action of G on the set of coset
s of H in G and hence a covering morphism p from, say, K to G, where K is a groupoid with vertex groups isomorphic to H. In this way, presentations of the group G can be "lifted" to presentations of the groupoid K, and this is a useful way of obtaining information about presentations of the subgroup H. For further information, see the books by Higgins and by Brown in the References.
Another useful fact is that the category of groupoids, unlike that of groups, is cartesian closed.
s.
These can be studied in terms of Lie algebroid
s, in analogy to the relation between Lie group
s and Lie algebra
s.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, especially in category theory
Category theory
Category theory is an area of study in mathematics that examines in an abstract way the properties of particular mathematical concepts, by formalising them as collections of objects and arrows , where these collections satisfy certain basic conditions...
and homotopy theory, a groupoid (less often Brandt groupoid or virtual group) generalises the notion of group
Group (mathematics)
In mathematics, a group is an algebraic structure consisting of a set together with an operation that combines any two of its elements to form a third element. To qualify as a group, the set and the operation must satisfy a few conditions called group axioms, namely closure, associativity, identity...
in several equivalent ways. A groupoid can be seen as a:
- GroupGroup (mathematics)In mathematics, a group is an algebraic structure consisting of a set together with an operation that combines any two of its elements to form a third element. To qualify as a group, the set and the operation must satisfy a few conditions called group axioms, namely closure, associativity, identity...
with a partial functionPartial functionIn mathematics, a partial function from X to Y is a function ƒ: X' → Y, where X' is a subset of X. It generalizes the concept of a function by not forcing f to map every element of X to an element of Y . If X' = X, then ƒ is called a total function and is equivalent to a function...
replacing the binary operationBinary operationIn mathematics, a binary operation is a calculation involving two operands, in other words, an operation whose arity is two. Examples include the familiar arithmetic operations of addition, subtraction, multiplication and division....
; - CategoryCategory theoryCategory theory is an area of study in mathematics that examines in an abstract way the properties of particular mathematical concepts, by formalising them as collections of objects and arrows , where these collections satisfy certain basic conditions...
in which every morphismMorphismIn mathematics, a morphism is an abstraction derived from structure-preserving mappings between two mathematical structures. The notion of morphism recurs in much of contemporary mathematics...
is invertible. A category of this sort can be viewed as augmented with a unary operationUnary operationIn mathematics, a unary operation is an operation with only one operand, i.e. a single input. Specifically, it is a functionf:\ A\to Awhere A is a set. In this case f is called a unary operation on A....
, called inverse by analogy with group theoryGroup theoryIn mathematics and abstract algebra, group theory studies the algebraic structures known as groups.The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces can all be seen as groups endowed with additional operations and...
. - Oriented graphGraph (mathematics)In mathematics, a graph is an abstract representation of a set of objects where some pairs of the objects are connected by links. The interconnected objects are represented by mathematical abstractions called vertices, and the links that connect some pairs of vertices are called edges...
Special cases include:
- SetoidSetoidIn mathematics, a setoid is a set equipped with an equivalence relation.Setoids are studied especially in proof theory and in type-theoretic foundations of mathematics. Often in mathematics, when one defines an equivalence relation on a set, one immediately forms the quotient set...
s, that is: sets which come with an equivalence relationEquivalence relationIn mathematics, an equivalence relation is a relation that, loosely speaking, partitions a set so that every element of the set is a member of one and only one cell of the partition. Two elements of the set are considered equivalent if and only if they are elements of the same cell...
; - G-sets, sets equipped with an actionGroup actionIn algebra and geometry, a group action is a way of describing symmetries of objects using groups. The essential elements of the object are described by a set, and the symmetries of the object are described by the symmetry group of this set, which consists of bijective transformations of the set...
of a group G.
Groupoids are often used to reason about geometrical objects such as manifold
Manifold
In mathematics , a manifold is a topological space that on a small enough scale resembles the Euclidean space of a specific dimension, called the dimension of the manifold....
s. Heinrich Brandt
Heinrich Brandt
Heinrich Brandt was a German mathematician who was the first to develop the concept of groupoid....
introduced groupoids implicitly via Brandt semigroups in 1926.
Algebraic
A groupoid is a set G with a unary operationUnary operation
In mathematics, a unary operation is an operation with only one operand, i.e. a single input. Specifically, it is a functionf:\ A\to Awhere A is a set. In this case f is called a unary operation on A....
and a partial functionPartial function
In mathematics, a partial function from X to Y is a function ƒ: X' → Y, where X' is a subset of X. It generalizes the concept of a function by not forcing f to map every element of X to an element of Y . If X' = X, then ƒ is called a total function and is equivalent to a function...
Here * is not a binary operationBinary operation
In mathematics, a binary operation is a calculation involving two operands, in other words, an operation whose arity is two. Examples include the familiar arithmetic operations of addition, subtraction, multiplication and division....
because it is not necessarily defined for all possible pairs of G-elements. The precise conditions under which * is defined are not articulated here and vary by situation.
and −1 have the following axiomatic properties. Let a, b, and c be elements of G. Then:
- AssociativityAssociativityIn mathematics, associativity is a property of some binary operations. It means that, within an expression containing two or more occurrences in a row of the same associative operator, the order in which the operations are performed does not matter as long as the sequence of the operands is not...
: If a * b and b * c are defined, then (a * b) * c and a * (b * c) are defined and equal. Conversely, if either of these last two expressions is defined, then so is the other (and again they are equal). - InverseInverse (mathematics)In many contexts in mathematics the term inverse indicates the opposite of something. This word and its derivatives are used widely in mathematics, as illustrated below....
: a−1 * a and a * a−1 are always defined. - IdentityIdentity (mathematics)In mathematics, the term identity has several different important meanings:*An identity is a relation which is tautologically true. This means that whatever the number or value may be, the answer stays the same. For example, algebraically, this occurs if an equation is satisfied for all values of...
: If a * b is defined, then a * b * b−1 = a, and a−1 * a * b = b. (The previous two axioms already show that these expressions are defined and unambiguous.)
In short:
- (a * b) * c = a * (b * c);
- ∀a ∃a−1 * a ∃a * a−1;
- a * b * b−1 = a and a−1 * a * b = b.
From these axioms, two easy and convenient properties follow:
- (a−1)−1 = a;
- If a * b is defined, then (a * b)−1 = b−1 * a−1.
Proof of first property: from 3. we obtain (a−1)−1=(a−1)−1 * a−1 * a = a. ✓
Proof of second property: since a * b and b * b−1 are defined, so is a * b * b−1=a. Therefore a * b * b−1 * a−1 is also defined. From 3. we obtain (a * b)−1 = (a * b)−1 * a*b * b−1 * a−1 = b−1 * a−1. ✓
Category theoretic
A groupoid is a small categoryCategory (mathematics)
In mathematics, a category is an algebraic structure that comprises "objects" that are linked by "arrows". A category has two basic properties: the ability to compose the arrows associatively and the existence of an identity arrow for each object. A simple example is the category of sets, whose...
in which every morphism
Morphism
In mathematics, a morphism is an abstraction derived from structure-preserving mappings between two mathematical structures. The notion of morphism recurs in much of contemporary mathematics...
is an isomorphism
Isomorphism
In abstract algebra, an isomorphism is a mapping between objects that shows a relationship between two properties or operations. If there exists an isomorphism between two structures, the two structures are said to be isomorphic. In a certain sense, isomorphic structures are...
, and hence invertible. More precisely, a groupoid G is:
- A set G0 of objects;
- For each pair of objects x and y in G0, there exists a (possibly empty) set G(x,y) of morphisms (or arrows) from x to y. We write f : x → y to indicate that f is an element of G(x,y).
The objects and morphisms have the properties:
- For every object x, there exists the element
of G(x,x); - For each triple of objects x, y, and z, there exists the functionFunction (mathematics)In mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can...
G(x,y)
G(y,z) → G(x,z). We write gf for 
, where f
G(x,y), and g
G(y,z); - There exists the functionFunction (mathematics)In mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can...
G(x,y) → G(y,x).
Moreover, if f : x → y, g : y → z, and h : z → w, then:
-
and 
; - (hg)f = h(gf);
-
and 
.
If f is an element of G(x,y) then x is called the source of f, written s(f), and y the target of f (written t(f)).
Comparing the definitions
The algebraic and category-theoretic definitions are equivalent, as follows. Given a groupoid in the category-theoretic sense, let G be the disjoint unionDisjoint union
In mathematics, the term disjoint union may refer to one of two different concepts:* In set theory, a disjoint union is a modified union operation that indexes the elements according to which set they originated in; disjoint sets have no element in common.* In probability theory , a disjoint union...
of all of the sets G(x,y) (i.e. the sets of morphisms from x to y). Then
and become partially defined operations on G, and will in fact be defined everywhere; so we define * to be and to be . Thus we have a groupoid in the algebraic sense. Explicit reference to G0 (and hence to ) can be dropped.Conversely, given a groupoid G in the algebraic sense, let G0 be the set of all elements of the form x * x−1 with x varying through G and define G(x*x -1,y*y -1) as the set of all elements f such that y * y -1 * f * x * x -1 exists. Given f∈G(x*x-1,y*y -1) and g∈G(y*y -1,z*z -1), their composite is defined as g * f ∈ G(x*x -1,z*z -1). To see this is well defined, observe that since z*z -1 * g * y*y -1 and y*y-1 * f * x*x-1 exist, so does z*z-1 * g * y*y-1 * y*y-1 * f * x*x -1 = z*z-1 * g*f * x*x -1. The identity morphism on x*x−1 is then x*x−1 itself, and the category-theoretic inverse of f is f-1.
Sets in the definitions above may be replaced with class
Class (set theory)
In set theory and its applications throughout mathematics, a class is a collection of sets which can be unambiguously defined by a property that all its members share. The precise definition of "class" depends on foundational context...
es, as is generally the case in category theory
Category theory
Category theory is an area of study in mathematics that examines in an abstract way the properties of particular mathematical concepts, by formalising them as collections of objects and arrows , where these collections satisfy certain basic conditions...
.
Vertex groups
Given a groupoid G, the vertex groups or isotropy groups or object groups in G are the subsets of the form G(x,x), where x is any object of G. It follows easily from the axioms above that these are indeed groups, as every pair of elements is composable and inverses are in the same vertex group.Category of groupoids
A subgroupoid is a subcategorySubcategory
In mathematics, a subcategory of a category C is a category S whose objects are objects in C and whose morphisms are morphisms in C with the same identities and composition of morphisms. Intuitively, a subcategory of C is a category obtained from C by "removing" some of its objects and...
that is itself a groupoid. A groupoid morphism is simply a functor between two (category-theoretic) groupoids. The category whose objects are groupoids and whose morphisms are groupoid morphisms is called the groupoid category, or the category of groupoids, denoted Grpd.
Particular kinds of morphisms of groupoids are of interest. A morphism $p: E \to B$ of groupoids is called a fibration if for each object $x$ of $E$ and each morphism $b$ of $B$ starting at $p(x)$ there is a morphism $e$ of $E$ starting at $x$ such that $p(e)=b$. A fibration is called a covering morphism if further such an $e$ is unique. The covering morphisms of groupoids are especially useful because they can be used to model covering map
Covering map
In mathematics, more specifically algebraic topology, a covering map is a continuous surjective function p from a topological space, C, to a topological space, X, such that each point in X has a neighbourhood evenly covered by p...
s of spaces. It is also true that the category of covering morphisms of a given groupoid $B$ is equivalent to the category of actions of the groupoid $B$ on sets.
Linear algebra
Given a field K, the corresponding general linear groupoid GL*(K) consists of all invertible matricesMatrix (mathematics)
In mathematics, a matrix is a rectangular array of numbers, symbols, or expressions. The individual items in a matrix are called its elements or entries. An example of a matrix with six elements isMatrices of the same size can be added or subtracted element by element...
whose entries range over K. Matrix multiplication
Matrix multiplication
In mathematics, matrix multiplication is a binary operation that takes a pair of matrices, and produces another matrix. If A is an n-by-m matrix and B is an m-by-p matrix, the result AB of their multiplication is an n-by-p matrix defined only if the number of columns m of the left matrix A is the...
interprets composition. If G = GL*(K), then the set of natural number
Natural number
In mathematics, the natural numbers are the ordinary whole numbers used for counting and ordering . These purposes are related to the linguistic notions of cardinal and ordinal numbers, respectively...
s is a proper subset of G0, since for each natural number
Natural number
In mathematics, the natural numbers are the ordinary whole numbers used for counting and ordering . These purposes are related to the linguistic notions of cardinal and ordinal numbers, respectively...
n, there is a corresponding identity matrix
Identity matrix
In linear algebra, the identity matrix or unit matrix of size n is the n×n square matrix with ones on the main diagonal and zeros elsewhere. It is denoted by In, or simply by I if the size is immaterial or can be trivially determined by the context...
of dimension n. G(m,n) is empty
Empty set
In mathematics, and more specifically set theory, the empty set is the unique set having no elements; its size or cardinality is zero. Some axiomatic set theories assure that the empty set exists by including an axiom of empty set; in other theories, its existence can be deduced...
unless m=n, in which case it is the set of all nxn invertible matrices.
Topology
Given a topological spaceTopological space
Topological spaces are mathematical structures that allow the formal definition of concepts such as convergence, connectedness, and continuity. They appear in virtually every branch of modern mathematics and are a central unifying notion...
X, let G0 be the set X. The morphisms from the point p to the point q are equivalence classes of continuous path
Path (topology)
In mathematics, a path in a topological space X is a continuous map f from the unit interval I = [0,1] to XThe initial point of the path is f and the terminal point is f. One often speaks of a "path from x to y" where x and y are the initial and terminal points of the path...
s from p to q, with two paths being equivalent if they are homotopic.
Two such morphisms are composed by first following the first path, then the second; the homotopy equivalence guarantees that this composition is associative. This groupoid is called the fundamental groupoid of X, denoted
(X). The usual fundamental group is then the vertex group for the point x.An important extension of this idea is to consider the fundamental groupoid
(X,A) where A is a set of "base points" and a subset of X. Here, one considers only paths whose endpoints belong to A. (X,A) is a sub-groupoid of (X). The set A may be chosen according to the geometry of the situation at hand.Equivalence relation
If X is a set with an equivalence relationEquivalence relation
In mathematics, an equivalence relation is a relation that, loosely speaking, partitions a set so that every element of the set is a member of one and only one cell of the partition. Two elements of the set are considered equivalent if and only if they are elements of the same cell...
denoted by infix
Infix
An infix is an affix inserted inside a word stem . It contrasts with adfix, a rare term for an affix attached to the end of a stem, such as a prefix or suffix.-Indonesian:...
, then a groupoid "representing" this equivalence relation can be formed as follows:
- The objects of the groupoid are the elements of X;
- For any two elements x and y in X, there is a single morphism from x to y if and only ifIf and only ifIn logic and related fields such as mathematics and philosophy, if and only if is a biconditional logical connective between statements....
x~y.
Group action
If the groupGroup (mathematics)
In mathematics, a group is an algebraic structure consisting of a set together with an operation that combines any two of its elements to form a third element. To qualify as a group, the set and the operation must satisfy a few conditions called group axioms, namely closure, associativity, identity...
G acts on the set X, then we can form the action groupoid representing this group action
Group action
In algebra and geometry, a group action is a way of describing symmetries of objects using groups. The essential elements of the object are described by a set, and the symmetries of the object are described by the symmetry group of this set, which consists of bijective transformations of the set...
as follows:
- The objects are the elements of X;
- For any two elements x and y in X, there is a morphismMorphismIn mathematics, a morphism is an abstraction derived from structure-preserving mappings between two mathematical structures. The notion of morphism recurs in much of contemporary mathematics...
from x to y corresponding to every element g of G such that gx = y; - CompositionFunction compositionIn mathematics, function composition is the application of one function to the results of another. For instance, the functions and can be composed by computing the output of g when it has an argument of f instead of x...
of morphisms interprets the binary operationBinary operationIn mathematics, a binary operation is a calculation involving two operands, in other words, an operation whose arity is two. Examples include the familiar arithmetic operations of addition, subtraction, multiplication and division....
of G.
More explicitly, the action groupoid is the set
with source and target maps s(g,x) = x and t(g,x) = gx. It is often denoted (or ). Multiplication (or composition) in the groupoid is then which is defined provided y=gx.For x in X, the vertex group consists of those (g,x) with gx = x, which is just the isotropy subgroup at x for the given action (which is why vertex groups are also called isotropy groups).
Another way to describe G-sets is the functor category
Functor category
In category theory, a branch of mathematics, the functors between two given categories form a category, where the objects are the functors and the morphisms are natural transformations between the functors...
, where is the groupoid (category) with one element and isomorphicIsomorphism
In abstract algebra, an isomorphism is a mapping between objects that shows a relationship between two properties or operations. If there exists an isomorphism between two structures, the two structures are said to be isomorphic. In a certain sense, isomorphic structures are...
to the group G. Indeed, every functor F of this category defines a set X=F
and for every g in G (i.e. for every morphism in ) induces a bijectionBijection
A bijection is a function giving an exact pairing of the elements of two sets. A bijection from the set X to the set Y has an inverse function from Y to X. If X and Y are finite sets, then the existence of a bijection means they have the same number of elements...
Fg : X→X. The categorical structure of the functor F assures us that F defines a G-action on the set X. The (unique) representable functor
Representable functor
In mathematics, particularly category theory, a representable functor is a functor of a special form from an arbitrary category into the category of sets. Such functors give representations of an abstract category in terms of known structures In mathematics, particularly category theory, a...
F :
→ is the Cayley representationCayley's theorem
In group theory, Cayley's theorem, named in honor of Arthur Cayley, states that every group G is isomorphic to a subgroup of the symmetric group acting on G...
of G. In fact, this functor is isomorphic to
and so sends to the set which is by definition the "set" G and the morphism g of (i.e. the element g of G) to the permutation Fg of the set G. We deduce from the Yoneda embedding that the group G is isomorphic to the group {Fg | g∈G}, a subgroupSubgroup
In group theory, given a group G under a binary operation *, a subset H of G is called a subgroup of G if H also forms a group under the operation *. More precisely, H is a subgroup of G if the restriction of * to H x H is a group operation on H...
of the group of permutation
Permutation group
In mathematics, a permutation group is a group G whose elements are permutations of a given set M, and whose group operation is the composition of permutations in G ; the relationship is often written as...
s of G.
Fifteen puzzle
The symmetries of the fifteen puzzle form a groupoid (not a group, as not all moves can be composed). This groupoid acts on configurations.Relation to groups
If a groupoid has only one object, then the set of its morphisms forms a group. Using the algebraic definition, such a groupoid is literally just a group. Many concepts of group theoryGroup theory
In mathematics and abstract algebra, group theory studies the algebraic structures known as groups.The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces can all be seen as groups endowed with additional operations and...
generalize to groupoids, with the notion of functor
Functor
In category theory, a branch of mathematics, a functor is a special type of mapping between categories. Functors can be thought of as homomorphisms between categories, or morphisms when in the category of small categories....
replacing that of group homomorphism
Group homomorphism
In mathematics, given two groups and , a group homomorphism from to is a function h : G → H such that for all u and v in G it holds that h = h \cdot h...
.
If x is an object of the groupoid G, then the set of all morphisms from x to x forms a group G(x). If there is a morphism f from x to y, then the groups G(x) and G(y) are isomorphic
Group isomorphism
In abstract algebra, a group isomorphism is a function between two groups that sets up a one-to-one correspondence between the elements of the groups in a way that respects the given group operations. If there exists an isomorphism between two groups, then the groups are called isomorphic...
, with an isomorphism given by the mapping
Map (mathematics)
In most of mathematics and in some related technical fields, the term mapping, usually shortened to map, is either a synonym for function, or denotes a particular kind of function which is important in that branch, or denotes something conceptually similar to a function.In graph theory, a map is a...
g → fgf −1.
Every connected groupoid (that is, one in which any two objects are connected by at least one morphism) is isomorphic to a groupoid of the following form. Pick a group G and a set (or class) X. Let the objects of the groupoid be the elements of X. For elements x and y of X, let the set of morphisms from x to y be G. Composition of morphisms is the group operation of G. If the groupoid is not connected, then it is isomorphic to a disjoint union
Disjoint union
In mathematics, the term disjoint union may refer to one of two different concepts:* In set theory, a disjoint union is a modified union operation that indexes the elements according to which set they originated in; disjoint sets have no element in common.* In probability theory , a disjoint union...
of groupoids of the above type (possibly with different groups G for each connected component). Thus any groupoid may be given (up to
Up to
In mathematics, the phrase "up to x" means "disregarding a possible difference in x".For instance, when calculating an indefinite integral, one could say that the solution is f "up to addition by a constant," meaning it differs from f, if at all, only by some constant.It indicates that...
isomorphism) by a set of ordered pair
Ordered pair
In mathematics, an ordered pair is a pair of mathematical objects. In the ordered pair , the object a is called the first entry, and the object b the second entry of the pair...
s (X,G).
Note that the isomorphism described above is not unique, and there is no natural choice. Choosing such an isomorphism for a connected groupoid essentially amounts to picking one object x0, a group isomorphism
Group isomorphism
In abstract algebra, a group isomorphism is a function between two groups that sets up a one-to-one correspondence between the elements of the groups in a way that respects the given group operations. If there exists an isomorphism between two groups, then the groups are called isomorphic...
h from G(x0) to G, and for each x other than x0, a morphism in G from x0 to x.
In category-theoretic terms, each connected component of a groupoid is equivalent (but not isomorphic) to a groupoid with a single object, that is, a single group. Thus any groupoid is equivalent to a multiset
Multiset
In mathematics, the notion of multiset is a generalization of the notion of set in which members are allowed to appear more than once...
of unrelated groups. In other words, for equivalence instead of isomorphism, one need not specify the sets X, only the groups G.
Consider the examples in the previous section. The general linear groupoid is both equivalent and isomorphic to the disjoint union of the various general linear group
General linear group
In mathematics, the general linear group of degree n is the set of n×n invertible matrices, together with the operation of ordinary matrix multiplication. This forms a group, because the product of two invertible matrices is again invertible, and the inverse of an invertible matrix is invertible...
s GLn(F). On the other hand:
- The fundamental groupoid of X is equivalent to the collection of the fundamental groupFundamental groupIn mathematics, more specifically algebraic topology, the fundamental group is a group associated to any given pointed topological space that provides a way of determining when two paths, starting and ending at a fixed base point, can be continuously deformed into each other...
s of each path-connected component of X, but an isomorphism requires specifying the set of points in each component; - The set X with the equivalence relation
is equivalent (as a groupoid) to one copy of the trivial groupTrivial groupIn mathematics, a trivial group is a group consisting of a single element. All such groups are isomorphic so one often speaks of the trivial group. The single element of the trivial group is the identity element so it usually denoted as such, 0, 1 or e depending on the context...
for each equivalence class, but an isomorphism requires specifying what each equivalence class is: - The set X equipped with an actionGroup actionIn algebra and geometry, a group action is a way of describing symmetries of objects using groups. The essential elements of the object are described by a set, and the symmetries of the object are described by the symmetry group of this set, which consists of bijective transformations of the set...
of the group G is equivalent (as a groupoid) to one copy of G for each orbit of the action, but an isomorphismIsomorphismIn abstract algebra, an isomorphism is a mapping between objects that shows a relationship between two properties or operations. If there exists an isomorphism between two structures, the two structures are said to be isomorphic. In a certain sense, isomorphic structures are...
requires specifying what set each orbit is.
The collapse of a groupoid into a mere collection of groups loses some information, even from a category-theoretic point of view, because it is not natural. Thus when groupoids arise in terms of other structures, as in the above examples, it can be helpful to maintain the full groupoid. Otherwise, one must choose a way to view each G(x) in terms of a single group, and this choice can be arbitrary. In our example from topology
Topology
Topology is a major area of mathematics concerned with properties that are preserved under continuous deformations of objects, such as deformations that involve stretching, but no tearing or gluing...
, you would have to make a coherent choice of paths (or equivalence classes of paths) from each point p to each point q in the same path-connected component.
As a more illuminating example, the classification of groupoids with one endomorphism
Endomorphism
In mathematics, an endomorphism is a morphism from a mathematical object to itself. For example, an endomorphism of a vector space V is a linear map ƒ: V → V, and an endomorphism of a group G is a group homomorphism ƒ: G → G. In general, we can talk about...
does not reduce to purely group theoretic considerations. This is analogous to the fact that the classification of vector space
Vector space
A vector space is a mathematical structure formed by a collection of vectors: objects that may be added together and multiplied by numbers, called scalars in this context. Scalars are often taken to be real numbers, but one may also consider vector spaces with scalar multiplication by complex...
s with one endomorphism is nontrivial.
Morphisms of groupoids come in more kinds than those of groups: we have, for example, fibration
Fibration
In topology, a branch of mathematics, a fibration is a generalization of the notion of a fiber bundle. A fiber bundle makes precise the idea of one topological space being "parameterized" by another topological space . A fibration is like a fiber bundle, except that the fibers need not be the same...
s, covering morphisms, universal morphisms, and quotient morphisms. Thus a subgroup H of a group G yields an action of G on the set of coset
Coset
In mathematics, if G is a group, and H is a subgroup of G, and g is an element of G, thenA coset is a left or right coset of some subgroup in G...
s of H in G and hence a covering morphism p from, say, K to G, where K is a groupoid with vertex groups isomorphic to H. In this way, presentations of the group G can be "lifted" to presentations of the groupoid K, and this is a useful way of obtaining information about presentations of the subgroup H. For further information, see the books by Higgins and by Brown in the References.
Another useful fact is that the category of groupoids, unlike that of groups, is cartesian closed.
Lie groupoids and Lie algebroids
When studying geometrical objects, the arising groupoids often carry some differentiable structure, turning them into Lie groupoidLie groupoid
In mathematics, a Lie groupoid is a groupoid where the set Ob of objects and the set Mor of morphisms are both manifolds, the source and target operationss,t : Mor \to Ob...
s.
These can be studied in terms of Lie algebroid
Lie algebroid
In mathematics, Lie algebroids serve the same role in the theory of Lie groupoids that Lie algebras serve in the theory of Lie groups: reducing global problems to infinitesimal ones...
s, in analogy to the relation between Lie group
Lie group
In mathematics, a Lie group is a group which is also a differentiable manifold, with the property that the group operations are compatible with the smooth structure...
s and Lie algebra
Lie algebra
In mathematics, a Lie algebra is an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable manifolds. Lie algebras were introduced to study the concept of infinitesimal transformations. The term "Lie algebra" was introduced by Hermann Weyl in the...
s.